A quantitative Selberg’s lemma
Tsachik Gelander
Northwestern University, Evanston, USARaz Slutsky
Weizmann Institute of Science, Rehovot, Israel; University of Oxford, Oxford, UK

Abstract
We show that an arithmetic lattice in a semi-simple Lie group contains a torsion-free subgroup of index where is the co-volume of the lattice. We prove that is polynomial in general and poly-logarithmic under the generalized Riemann hypothesis (GRH). We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order .
Cite this article
Tsachik Gelander, Raz Slutsky, A quantitative Selberg’s lemma. Groups Geom. Dyn. (2025), published online first
DOI 10.4171/GGD/865